Write an equation of a Ferris wheel

Question

Problem: You are about to begin your ride at the very bottom of the ferris wheel, and rotate counter-clockwise from there. Write a possible equation of the graph with these details: Diameter is 228ft, Speed: 14 minutes per revolution, initial height off ground: 8 ft, Max height: 236

b. How high off the ground are you going to be after 5 minutes?

c. How high off the ground are you going to be after 10 minutes?

d.How high off the ground are you going to be after 15 minutes?

e.If you stay on the ferris wheel for a really long time , list the first 3 times where you will be exactly at 30 ft. off the ground?

 

f.Using your equation , find the coordinate of the point where t=-20 minutes. Describe what those coordiantes represent in the context of the problem.

Answer 1

Swathi,

 

This is circular (rotational) motion with a constant angular velocity ω = 1 revolutions / 14 minutes, where each revolution is 2π radians, so ω = 2π radians / 14 minutes.

 

θ(t) - θ(t=0) = ωt

 

The height in terms of the angle is the side of the triangle opposite the angle. The hypotenuse of the triangle is the radius of the ferris wheel. Since sin θ(t) = opposite/hypotenuse, height(t) = opposite = hypotenuse * sin θ(t) = radius * sin θ(t).

 

Initially, the very bottom of the circle (ferris wheel) is 8 feet off the ground. So, you have to add 8 ft to the height(t):

 

height(t) = radius * sin θ(t) + 8 ft

 

Since the sine is zero at the right of the center of the circle at the horizon, θ = θ(t = 0) is a quarter revolution or 2π/4 radians back: θ(t = 0) = -π/2

 

Since the radius is the diameter, 228 ft, divided by two, height(t) = 114 ft * sin[(2π radians / 14 minutes) * t -π/2] + 8 ft.

 

In b, c, and d, substitute for t and calculate height(t).

 

In e, set hieght(t) = 30 ft and solve for t in the argument of the sine.

 

Michael.